It boils down to orbital hybridization, which is related to the amount of negative charge centers an atom has. A lone carbon atom has 4 electrons available for bonding, so if it bonds to 4 hydrogen atoms it has 4 negative charge centers around it. In this case, because it has 4 negative charge centers we say the carbon atom is $sp^3$ hybridized, meaning that the s orbital and the three p orbitals have mixed to create 4 $sp^3$ hybrid orbitals in a tetrahedral shape around the carbon atom. These hybrid orbitals bond with the hydrogen by overlapping with its orbital/electron cloud, which is known as a sigma bond, or $\sigma$ bond. $\sigma$ bonds are any bonds formed by the overlapping of hybridized orbitals, and their electron density is on the bond axis.
However, in a double bond, such as $C_2H_4$, there are only 3 negative charge centers around each carbon atom, and thus the orbitals are only $sp^2$ hybridized, which means that there is still an unhybridized p orbital around each carbon atom. While the hybridized orbitals overlap and form a sigma bond, the unhybridized p orbitals also form a bond (the second bond in the double bond). Because these bonds are formed by two unhybridized orbitals, they are known as pi bonds, or $\pi$ bonds.
Essentially, every covalent bonding between two atoms has one sigma bond, but if it's a double bond it also has a pi bond, and if it's a triple bond it has two pi bonds along with the sigma bond.
The non-theoretical definition would be if there are two or three bonds (one sigma and one or two pi bonds) between two atoms, then it is a double or triple bond respectively. If an atom is bonded to two other atoms by a sigma bond for each, then it has two single bonds.